The most successful way I've seen of killing a tree is to girdle it. It is not immediate, but doesn't involve poison or frequent attention. This will be harder to do on a tree that has previously been cut down and has many stems. Once you cut the top of the tree off, hormones in the tree trigger it to grow lots of shoots. So, if possible, avoid that step for now. Cut through the bark (being careful to not cut the wood much) and peel off at least 6 inches of bark around the full circumference of the tree. Be sure you aren't leaving the inner bark (even a small strip in a fold of the trunk can sustain the tree). The spring when the bark slips is the easiest time to do this. If you don't damage the wood and only cut the bark, the tree won't even know it is dying. It won't be able to send food between the leaves and roots. It won't store any energy to make it through the winter and it won't come back the next year. It will stay green through the summer, so it will look like you haven't done anything, but be patient.